I was thinking on the twin prime conjecture, that there are an infinite number of twin primes... I came up with a proof. I have to think that it is incomplete or wrong, because many great minds have thought on this previously. I can't see the issue, so I thought I would raise it in a broader forum. What is wrong with this proof?
1) A number n is prime iff n mod p is non zero for every prime number 1 < p < n
This is easy to prove from the definition of a prime number and mod. It just says that n is not equally divisible by another prime, making it prime.
Let $p_n$ = the nth prime $p_0 = 1$, $p_1 = 2$ ...
Consider $N_n = \Pi_0^n p_n$.
It is easy to see that $N_n \mod p_j = 0 $ for all primes $0 < j < n$
$(N_n + 1) \mod p_j = 1 $ for all primes $0 < j < n$, and therefore, from #1 must be prime
$(N_n - 1) \mod p_j = (p_j - 1) $ for primes $0 < j < n$, and therefore, from #1 must be prime as well
The set $p_n$ has infinite members (as shown by Euclid) , so there are infinite $N_n$. Therefore there are an infinite set of twin prime numbers $(N_n-1,N_n+1)$ .